L. Sciavicco and B. Siciliano

Modelling and Controlof Robot ManipulatorsSecond Edition

With 190 Figures

Springer

Table of Contents

1. Introduction 11.1 Robotics 11.2 Industrial Robot 31.3 Manipulator Structures 61.4 Modelling and Control of Robot Manipulators 14

1.4.1 Modelling 151.4.2 Control 15

1.5 Bibliographical Reference Texts 16

2. Kinematics 212.1 Position and Orientation of a Rigid Body 212.2 Rotation Matrix 22

2.2.1 Elementary Rotations 232.2.2 Representation of a Vector 242.2.3 Rotation of a Vector 25

2.3 Composition of Rotation Matrices 262.4 Euler Angles 30

2.4.1 ZYZ Angles 302.4.2 Roll-Pitch-Yaw Angles 32

2.5 Angle and Axis 332.6 Unit Quaternion 352.7 Homogeneous Transformations 372.8 Direct Kinematics 39

2.8.1 Open Chain 412.8.2 Denavit-Hartenberg Convention 422.8.3 Closed Chain 46

2.9 Kinematics of Typical Manipulator Structures 492.9.1 Three-link Planar Arm 492.9.2 Parallelogram Arm 512.9.3 Spherical Arm 532.9.4 Anthropomorphic Arm 542.9.5 Spherical Wrist 552.9.6 Stanford Manipulator 572.9.7 Anthropomorphic Arm with Spherical Wrist 58

2.10 Joint Space and Operational Space 592.10.1 Workspace 61

xx Table of Contents

2.10.2 Kinematic Redundancy 632.11 Kinematic Calibration 642.12 Inverse Kinematics Problem 66

2.12.1 Solution of Three-link Planar Arm 672.12.2 Solution of Manipulators with Spherical Wrist 702.12.3 Solution of Spherical Arm 712.12.4 Solution of Anthropomorphic Arm 722.12.5 Solution of Spherical Wrist 73Problems 74Bibliography 76

3. Differential Kinematics and Statics* 793.1 Geometric Jacobian 79

3.1.1 Derivative of a Rotation Matrix 803.1.2 Link Velocity 823.1.3 Jacobian Computation 84

3.2 Jacobian of Typical Manipulator Structures 873.2.1 Three-link Planar Arm 873.2.2 Anthropomorphic Arm 883.2.3 Stanford Manipulator 89

3.3 Kinematic Singularities 893.3.1 Singularity Decoupling 913.3.2 Wrist Singularities 923.3.3 Arm Singularities 92

3.4 Analysis of Redundancy 943.5 Differential Kinematics Inversion 96

3.5.1 Redundant Manipulators 973.5.2 Kinematic Singularities 100

3.6 Analytical Jacobian 1003.7 Inverse Kinematics Algorithms 104

3.7.1 Jacobian (Pseudo-)Inverse 1053.7.2 Jacobian Transpose 1063.7.3 Orientation Error 1093.7.4 A Comparison Between Inverse Kinematics Algorithms .. 112

3.8 Statics 1163.8.1 Kineto-statics Duality 1183.8.2 Velocity and Force Transformation 1193.8.3 Closed Chain 120

3.9 Manipulability Ellipsoids 121Problems 127Bibliography 128

4. Dynamics 1314.1 Lagrange Formulation 131

4.1.1 Computation of Kinetic Energy 1334.1.2 Computation of Potential Energy 1394.1.3 Equations of Motion 140